Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 14.0s

Alternatives: 22

Speedup: 1.0×

• could not determine a ground truth

  1. NdChar = 1.922781407509087e-8
  2. Ec = 2.5027613892890587e+24
  3. Vef = -4.6129948870541765e-133
  4. EDonor = 9.216719160556246e-234
  5. mu = 1.3978069162952202e+250
  6. KbT = -1.7658302214871299e+190
  7. NaChar = -5.5756306115181404e-8
  8. Ev = 3.284225668331604e+265
  9. EAccept = 2.051287217992295e+21

Specification

Math

\[\begin{array}{l} \\ \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (+
  (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
  (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    code = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) + -mu) / kbt))))
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT))))

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) + Float64(-mu)) / KbT)))))
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] + (-mu)), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (+
  (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
  (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    code = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) + -mu) / kbt))))
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT))))

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) + Float64(-mu)) / KbT)))))
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] + (-mu)), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} - \frac{NdChar}{-1 - {\left(e^{-1}\right)}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (-
  (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT))))
  (/ NdChar (- -1.0 (pow (exp -1.0) (/ (- (- (- Ec Vef) EDonor) mu) KbT))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)))) - (NdChar / (-1.0 - pow(exp(-1.0), ((((Ec - Vef) - EDonor) - mu) / KbT))));
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    code = (nachar / (1.0d0 + exp((((eaccept + (ev + vef)) - mu) / kbt)))) - (ndchar / ((-1.0d0) - (exp((-1.0d0)) ** ((((ec - vef) - edonor) - mu) / kbt))))
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NaChar / (1.0 + Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)))) - (NdChar / (-1.0 - Math.pow(Math.exp(-1.0), ((((Ec - Vef) - EDonor) - mu) / KbT))));
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return (NaChar / (1.0 + math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)))) - (NdChar / (-1.0 - math.pow(math.exp(-1.0), ((((Ec - Vef) - EDonor) - mu) / KbT))))

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)))) - Float64(NdChar / Float64(-1.0 - (exp(-1.0) ^ Float64(Float64(Float64(Float64(Ec - Vef) - EDonor) - mu) / KbT)))))
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = (NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)))) - (NdChar / (-1.0 - (exp(-1.0) ^ ((((Ec - Vef) - EDonor) - mu) / KbT))));
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Power[N[Exp[-1.0], $MachinePrecision], N[(N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \frac{NdChar}{1 + \color{blue}{e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. lift-neg.f64 N/A

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   4. distribute-frac-neg N/A

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\mathsf{neg}\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   5. neg-mul-1 N/A

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{-1 \cdot \frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   6. exp-prod N/A

      \[\leadsto \frac{NdChar}{1 + \color{blue}{{\left(e^{-1}\right)}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   7. lower-pow.f64 N/A

      \[\leadsto \frac{NdChar}{1 + \color{blue}{{\left(e^{-1}\right)}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   8. lower-exp.f64 N/A

      \[\leadsto \frac{NdChar}{1 + {\color{blue}{\left(e^{-1}\right)}}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   9. lower-/.f64 100.0

      \[\leadsto \frac{NdChar}{1 + {\left(e^{-1}\right)}^{\color{blue}{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

4. Applied rewrites 100.0%

   \[\leadsto \frac{NdChar}{1 + \color{blue}{{\left(e^{-1}\right)}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

5. Final simplification 100.0%

   \[\leadsto \frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} - \frac{NdChar}{-1 - {\left(e^{-1}\right)}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}} \]
6. Add Preprocessing

Alternative 2: 77.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}\\ t_1 := \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} - \frac{NdChar}{-1 - t\_0}\\ t_2 := e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\\ t_3 := \frac{NdChar}{1 + t\_0} - \frac{NaChar}{-1 - t\_2}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-250}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}}}\\ \mathbf{elif}\;t\_3 \leq 10^{-12}:\\ \;\;\;\;\frac{NaChar}{1 + t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT)))
        (t_1
         (- (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) (/ NdChar (- -1.0 t_0))))
        (t_2 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))
        (t_3 (- (/ NdChar (+ 1.0 t_0)) (/ NaChar (- -1.0 t_2)))))
   (if (<= t_3 -2e-250)
     t_1
     (if (<= t_3 5e-78)
       (/ NdChar (+ 1.0 (exp (/ (- (+ (+ mu Vef) EDonor) Ec) KbT))))
       (if (<= t_3 1e-12) (/ NaChar (+ 1.0 t_2)) t_1)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp(((mu - ((Ec - Vef) - EDonor)) / KbT));
	double t_1 = (NaChar / (1.0 + exp((EAccept / KbT)))) - (NdChar / (-1.0 - t_0));
	double t_2 = exp((((EAccept + (Ev + Vef)) - mu) / KbT));
	double t_3 = (NdChar / (1.0 + t_0)) - (NaChar / (-1.0 - t_2));
	double tmp;
	if (t_3 <= -2e-250) {
		tmp = t_1;
	} else if (t_3 <= 5e-78) {
		tmp = NdChar / (1.0 + exp(((((mu + Vef) + EDonor) - Ec) / KbT)));
	} else if (t_3 <= 1e-12) {
		tmp = NaChar / (1.0 + t_2);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = exp(((mu - ((ec - vef) - edonor)) / kbt))
    t_1 = (nachar / (1.0d0 + exp((eaccept / kbt)))) - (ndchar / ((-1.0d0) - t_0))
    t_2 = exp((((eaccept + (ev + vef)) - mu) / kbt))
    t_3 = (ndchar / (1.0d0 + t_0)) - (nachar / ((-1.0d0) - t_2))
    if (t_3 <= (-2d-250)) then
        tmp = t_1
    else if (t_3 <= 5d-78) then
        tmp = ndchar / (1.0d0 + exp(((((mu + vef) + edonor) - ec) / kbt)))
    else if (t_3 <= 1d-12) then
        tmp = nachar / (1.0d0 + t_2)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT));
	double t_1 = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) - (NdChar / (-1.0 - t_0));
	double t_2 = Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT));
	double t_3 = (NdChar / (1.0 + t_0)) - (NaChar / (-1.0 - t_2));
	double tmp;
	if (t_3 <= -2e-250) {
		tmp = t_1;
	} else if (t_3 <= 5e-78) {
		tmp = NdChar / (1.0 + Math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)));
	} else if (t_3 <= 1e-12) {
		tmp = NaChar / (1.0 + t_2);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT))
	t_1 = (NaChar / (1.0 + math.exp((EAccept / KbT)))) - (NdChar / (-1.0 - t_0))
	t_2 = math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))
	t_3 = (NdChar / (1.0 + t_0)) - (NaChar / (-1.0 - t_2))
	tmp = 0
	if t_3 <= -2e-250:
		tmp = t_1
	elif t_3 <= 5e-78:
		tmp = NdChar / (1.0 + math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)))
	elif t_3 <= 1e-12:
		tmp = NaChar / (1.0 + t_2)
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT))
	t_1 = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) - Float64(NdChar / Float64(-1.0 - t_0)))
	t_2 = exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))
	t_3 = Float64(Float64(NdChar / Float64(1.0 + t_0)) - Float64(NaChar / Float64(-1.0 - t_2)))
	tmp = 0.0
	if (t_3 <= -2e-250)
		tmp = t_1;
	elseif (t_3 <= 5e-78)
		tmp = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT))));
	elseif (t_3 <= 1e-12)
		tmp = Float64(NaChar / Float64(1.0 + t_2));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((mu - ((Ec - Vef) - EDonor)) / KbT));
	t_1 = (NaChar / (1.0 + exp((EAccept / KbT)))) - (NdChar / (-1.0 - t_0));
	t_2 = exp((((EAccept + (Ev + Vef)) - mu) / KbT));
	t_3 = (NdChar / (1.0 + t_0)) - (NaChar / (-1.0 - t_2));
	tmp = 0.0;
	if (t_3 <= -2e-250)
		tmp = t_1;
	elseif (t_3 <= 5e-78)
		tmp = NdChar / (1.0 + exp(((((mu + Vef) + EDonor) - Ec) / KbT)));
	elseif (t_3 <= 1e-12)
		tmp = NaChar / (1.0 + t_2);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(NdChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-250], t$95$1, If[LessEqual[t$95$3, 5e-78], N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e-12], N[(NaChar / N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2.0000000000000001e-250 or 9.9999999999999998e-13 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in EAccept around inf

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 80.9

         \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]

   5. Applied rewrites 80.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]

   if -2.0000000000000001e-250 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.9999999999999996e-78

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 6.8

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 6.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

   6. Taylor expanded in NdChar around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NaChar} \]
   7. Step-by-step derivation

   8. Applied rewrites 7.1%

      \[\leadsto 0.5 \cdot \color{blue}{NaChar} \]

   9. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

       2. +-commutative N/A

          \[\leadsto \frac{NdChar}{\color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}} + 1}} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{NdChar}{\color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}} + 1}} \]

       4. lower-exp.f64 N/A

          \[\leadsto \frac{NdChar}{\color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}} + 1} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{NdChar}{e^{\color{blue}{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}} + 1} \]

       6. lower--.f64 N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\color{blue}{\left(EDonor + \left(Vef + mu\right)\right) - Ec}}{KbT}} + 1} \]

       7. +-commutative N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\color{blue}{\left(\left(Vef + mu\right) + EDonor\right)} - Ec}{KbT}} + 1} \]

       8. lower-+.f64 N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\color{blue}{\left(\left(Vef + mu\right) + EDonor\right)} - Ec}{KbT}} + 1} \]

       9. +-commutative N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\left(\color{blue}{\left(mu + Vef\right)} + EDonor\right) - Ec}{KbT}} + 1} \]

       10. lower-+.f64 86.8

           \[\leadsto \frac{NdChar}{e^{\frac{\left(\color{blue}{\left(mu + Vef\right)} + EDonor\right) - Ec}{KbT}} + 1} \]

   11. Applied rewrites 86.8%

       \[\leadsto \color{blue}{\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + 1}} \]

   if 4.9999999999999996e-78 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 9.9999999999999998e-13

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{NaChar}{\color{blue}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{NaChar}{\color{blue}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1}} \]

      4. lower-exp.f64 N/A

         \[\leadsto \frac{NaChar}{\color{blue}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} + 1} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{NaChar}{e^{\color{blue}{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} + 1} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{NaChar}{e^{\frac{\color{blue}{\left(EAccept + \left(Ev + Vef\right)\right) - mu}}{KbT}} + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{NaChar}{e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right)} - mu}{KbT}} + 1} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{NaChar}{e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right)} - mu}{KbT}} + 1} \]

      9. lower-+.f64 80.8

         \[\leadsto \frac{NaChar}{e^{\frac{\left(\color{blue}{\left(Ev + Vef\right)} + EAccept\right) - mu}{KbT}} + 1} \]

   5. Applied rewrites 80.8%

      \[\leadsto \color{blue}{\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq -2 \cdot 10^{-250}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 10^{-12}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ t_1 := \frac{NaChar}{2} + t\_0\\ t_2 := e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\\ t_3 := t\_0 - \frac{NaChar}{-1 - t\_2}\\ \mathbf{if}\;t\_3 \leq -4 \cdot 10^{-228}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}}}\\ \mathbf{elif}\;t\_3 \leq 10^{-12}:\\ \;\;\;\;\frac{NaChar}{1 + t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT)))))
        (t_1 (+ (/ NaChar 2.0) t_0))
        (t_2 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))
        (t_3 (- t_0 (/ NaChar (- -1.0 t_2)))))
   (if (<= t_3 -4e-228)
     t_1
     (if (<= t_3 5e-78)
       (/ NdChar (+ 1.0 (exp (/ (- (+ (+ mu Vef) EDonor) Ec) KbT))))
       (if (<= t_3 1e-12) (/ NaChar (+ 1.0 t_2)) t_1)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)));
	double t_1 = (NaChar / 2.0) + t_0;
	double t_2 = exp((((EAccept + (Ev + Vef)) - mu) / KbT));
	double t_3 = t_0 - (NaChar / (-1.0 - t_2));
	double tmp;
	if (t_3 <= -4e-228) {
		tmp = t_1;
	} else if (t_3 <= 5e-78) {
		tmp = NdChar / (1.0 + exp(((((mu + Vef) + EDonor) - Ec) / KbT)));
	} else if (t_3 <= 1e-12) {
		tmp = NaChar / (1.0 + t_2);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp(((mu - ((ec - vef) - edonor)) / kbt)))
    t_1 = (nachar / 2.0d0) + t_0
    t_2 = exp((((eaccept + (ev + vef)) - mu) / kbt))
    t_3 = t_0 - (nachar / ((-1.0d0) - t_2))
    if (t_3 <= (-4d-228)) then
        tmp = t_1
    else if (t_3 <= 5d-78) then
        tmp = ndchar / (1.0d0 + exp(((((mu + vef) + edonor) - ec) / kbt)))
    else if (t_3 <= 1d-12) then
        tmp = nachar / (1.0d0 + t_2)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)));
	double t_1 = (NaChar / 2.0) + t_0;
	double t_2 = Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT));
	double t_3 = t_0 - (NaChar / (-1.0 - t_2));
	double tmp;
	if (t_3 <= -4e-228) {
		tmp = t_1;
	} else if (t_3 <= 5e-78) {
		tmp = NdChar / (1.0 + Math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)));
	} else if (t_3 <= 1e-12) {
		tmp = NaChar / (1.0 + t_2);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))
	t_1 = (NaChar / 2.0) + t_0
	t_2 = math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))
	t_3 = t_0 - (NaChar / (-1.0 - t_2))
	tmp = 0
	if t_3 <= -4e-228:
		tmp = t_1
	elif t_3 <= 5e-78:
		tmp = NdChar / (1.0 + math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)))
	elif t_3 <= 1e-12:
		tmp = NaChar / (1.0 + t_2)
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT))))
	t_1 = Float64(Float64(NaChar / 2.0) + t_0)
	t_2 = exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))
	t_3 = Float64(t_0 - Float64(NaChar / Float64(-1.0 - t_2)))
	tmp = 0.0
	if (t_3 <= -4e-228)
		tmp = t_1;
	elseif (t_3 <= 5e-78)
		tmp = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT))));
	elseif (t_3 <= 1e-12)
		tmp = Float64(NaChar / Float64(1.0 + t_2));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)));
	t_1 = (NaChar / 2.0) + t_0;
	t_2 = exp((((EAccept + (Ev + Vef)) - mu) / KbT));
	t_3 = t_0 - (NaChar / (-1.0 - t_2));
	tmp = 0.0;
	if (t_3 <= -4e-228)
		tmp = t_1;
	elseif (t_3 <= 5e-78)
		tmp = NdChar / (1.0 + exp(((((mu + Vef) + EDonor) - Ec) / KbT)));
	elseif (t_3 <= 1e-12)
		tmp = NaChar / (1.0 + t_2);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / 2.0), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 - N[(NaChar / N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -4e-228], t$95$1, If[LessEqual[t$95$3, 5e-78], N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e-12], N[(NaChar / N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4.00000000000000013e-228 or 9.9999999999999998e-13 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{\color{blue}{\left(2 + \left(\frac{EAccept}{KbT} + \left(\frac{Ev}{KbT} + \frac{Vef}{KbT}\right)\right)\right) - \frac{mu}{KbT}}} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{\color{blue}{\left(2 + \left(\frac{EAccept}{KbT} + \left(\frac{Ev}{KbT} + \frac{Vef}{KbT}\right)\right)\right) - \frac{mu}{KbT}}} \]

      2. associate-+r+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{\color{blue}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Ev}{KbT} + \frac{Vef}{KbT}\right)\right)} - \frac{mu}{KbT}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{\color{blue}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Ev}{KbT} + \frac{Vef}{KbT}\right)\right)} - \frac{mu}{KbT}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{\left(\color{blue}{\left(2 + \frac{EAccept}{KbT}\right)} + \left(\frac{Ev}{KbT} + \frac{Vef}{KbT}\right)\right) - \frac{mu}{KbT}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{\left(\left(2 + \color{blue}{\frac{EAccept}{KbT}}\right) + \left(\frac{Ev}{KbT} + \frac{Vef}{KbT}\right)\right) - \frac{mu}{KbT}} \]

      6. +-commutative N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \color{blue}{\left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)}\right) - \frac{mu}{KbT}} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \color{blue}{\left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)}\right) - \frac{mu}{KbT}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\color{blue}{\frac{Vef}{KbT}} + \frac{Ev}{KbT}\right)\right) - \frac{mu}{KbT}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Vef}{KbT} + \color{blue}{\frac{Ev}{KbT}}\right)\right) - \frac{mu}{KbT}} \]

      10. lower-/.f64 64.6

          \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) - \color{blue}{\frac{mu}{KbT}}} \]

   5. Applied rewrites 64.6%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{\color{blue}{\left(\left(2 + \frac{EAccept}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) - \frac{mu}{KbT}}} \]

   6. Taylor expanded in KbT around inf

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 67.7%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{2} \]

   if -4.00000000000000013e-228 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.9999999999999996e-78

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 6.6

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 6.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

   6. Taylor expanded in NdChar around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NaChar} \]
   7. Step-by-step derivation

   8. Applied rewrites 7.4%

      \[\leadsto 0.5 \cdot \color{blue}{NaChar} \]

   9. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

       2. +-commutative N/A

          \[\leadsto \frac{NdChar}{\color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}} + 1}} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{NdChar}{\color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}} + 1}} \]

       4. lower-exp.f64 N/A

          \[\leadsto \frac{NdChar}{\color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}} + 1} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{NdChar}{e^{\color{blue}{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}} + 1} \]

       6. lower--.f64 N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\color{blue}{\left(EDonor + \left(Vef + mu\right)\right) - Ec}}{KbT}} + 1} \]

       7. +-commutative N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\color{blue}{\left(\left(Vef + mu\right) + EDonor\right)} - Ec}{KbT}} + 1} \]

       8. lower-+.f64 N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\color{blue}{\left(\left(Vef + mu\right) + EDonor\right)} - Ec}{KbT}} + 1} \]

       9. +-commutative N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\left(\color{blue}{\left(mu + Vef\right)} + EDonor\right) - Ec}{KbT}} + 1} \]

       10. lower-+.f64 85.1

           \[\leadsto \frac{NdChar}{e^{\frac{\left(\color{blue}{\left(mu + Vef\right)} + EDonor\right) - Ec}{KbT}} + 1} \]

   11. Applied rewrites 85.1%

       \[\leadsto \color{blue}{\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + 1}} \]

   if 4.9999999999999996e-78 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 9.9999999999999998e-13

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{NaChar}{\color{blue}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{NaChar}{\color{blue}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1}} \]

      4. lower-exp.f64 N/A

         \[\leadsto \frac{NaChar}{\color{blue}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} + 1} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{NaChar}{e^{\color{blue}{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} + 1} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{NaChar}{e^{\frac{\color{blue}{\left(EAccept + \left(Ev + Vef\right)\right) - mu}}{KbT}} + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{NaChar}{e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right)} - mu}{KbT}} + 1} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{NaChar}{e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right)} - mu}{KbT}} + 1} \]

      9. lower-+.f64 80.8

         \[\leadsto \frac{NaChar}{e^{\frac{\left(\color{blue}{\left(Ev + Vef\right)} + EAccept\right) - mu}{KbT}} + 1} \]

   5. Applied rewrites 80.8%

      \[\leadsto \color{blue}{\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq -4 \cdot 10^{-228}:\\ \;\;\;\;\frac{NaChar}{2} + \frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 10^{-12}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2} + \frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 35.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NaChar + NdChar\right) \cdot 0.5\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-214}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{1}{\frac{1 + \frac{\frac{NdChar \cdot NdChar}{NaChar} - NdChar}{NaChar}}{NaChar}} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-103}:\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ NaChar NdChar) 0.5))
        (t_1
         (-
          (/ NdChar (+ 1.0 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT))))
          (/ NaChar (- -1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))))))
   (if (<= t_1 -1e-214)
     t_0
     (if (<= t_1 0.0)
       (*
        (/
         1.0
         (/ (+ 1.0 (/ (- (/ (* NdChar NdChar) NaChar) NdChar) NaChar)) NaChar))
        0.5)
       (if (<= t_1 5e-103) (* 0.5 NdChar) t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	double tmp;
	if (t_1 <= -1e-214) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (1.0 / ((1.0 + ((((NdChar * NdChar) / NaChar) - NdChar) / NaChar)) / NaChar)) * 0.5;
	} else if (t_1 <= 5e-103) {
		tmp = 0.5 * NdChar;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (nachar + ndchar) * 0.5d0
    t_1 = (ndchar / (1.0d0 + exp(((mu - ((ec - vef) - edonor)) / kbt)))) - (nachar / ((-1.0d0) - exp((((eaccept + (ev + vef)) - mu) / kbt))))
    if (t_1 <= (-1d-214)) then
        tmp = t_0
    else if (t_1 <= 0.0d0) then
        tmp = (1.0d0 / ((1.0d0 + ((((ndchar * ndchar) / nachar) - ndchar) / nachar)) / nachar)) * 0.5d0
    else if (t_1 <= 5d-103) then
        tmp = 0.5d0 * ndchar
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (1.0 + Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	double tmp;
	if (t_1 <= -1e-214) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (1.0 / ((1.0 + ((((NdChar * NdChar) / NaChar) - NdChar) / NaChar)) / NaChar)) * 0.5;
	} else if (t_1 <= 5e-103) {
		tmp = 0.5 * NdChar;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar + NdChar) * 0.5
	t_1 = (NdChar / (1.0 + math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))))
	tmp = 0
	if t_1 <= -1e-214:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = (1.0 / ((1.0 + ((((NdChar * NdChar) / NaChar) - NdChar) / NaChar)) / NaChar)) * 0.5
	elif t_1 <= 5e-103:
		tmp = 0.5 * NdChar
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar + NdChar) * 0.5)
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)))) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)))))
	tmp = 0.0
	if (t_1 <= -1e-214)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(1.0 / Float64(Float64(1.0 + Float64(Float64(Float64(Float64(NdChar * NdChar) / NaChar) - NdChar) / NaChar)) / NaChar)) * 0.5);
	elseif (t_1 <= 5e-103)
		tmp = Float64(0.5 * NdChar);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NaChar + NdChar) * 0.5;
	t_1 = (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	tmp = 0.0;
	if (t_1 <= -1e-214)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = (1.0 / ((1.0 + ((((NdChar * NdChar) / NaChar) - NdChar) / NaChar)) / NaChar)) * 0.5;
	elseif (t_1 <= 5e-103)
		tmp = 0.5 * NdChar;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar + NdChar), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-214], t$95$0, If[LessEqual[t$95$1, 0.0], N[(N[(1.0 / N[(N[(1.0 + N[(N[(N[(N[(NdChar * NdChar), $MachinePrecision] / NaChar), $MachinePrecision] - NdChar), $MachinePrecision] / NaChar), $MachinePrecision]), $MachinePrecision] / NaChar), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 5e-103], N[(0.5 * NdChar), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -9.99999999999999913e-215 or 4.99999999999999966e-103 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 38.7

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 38.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

   if -9.99999999999999913e-215 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 2.5

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 2.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 3.9%

      \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{NdChar - NaChar}{\left(NaChar + NdChar\right) \cdot \left(NdChar - NaChar\right)}}} \]

   8. Taylor expanded in NdChar around inf

      \[\leadsto \frac{1}{2} \cdot \frac{1}{\frac{1}{\color{blue}{NdChar}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 4.0%

       \[\leadsto 0.5 \cdot \frac{1}{\frac{1}{\color{blue}{NdChar}}} \]

   11. Taylor expanded in NaChar around -inf

       \[\leadsto \frac{1}{2} \cdot \frac{1}{-1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{{NdChar}^{2}}{NaChar} - NdChar}{NaChar} - 1}{NaChar}}} \]
   12. Step-by-step derivation

   13. Applied rewrites 39.4%

       \[\leadsto 0.5 \cdot \frac{1}{\frac{\frac{\frac{NdChar \cdot NdChar}{NaChar} - NdChar}{NaChar} + 1}{\color{blue}{NaChar}}} \]

   if 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999966e-103

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 9.1

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 9.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

   6. Taylor expanded in NdChar around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NaChar} \]
   7. Step-by-step derivation

   8. Applied rewrites 13.0%

      \[\leadsto 0.5 \cdot \color{blue}{NaChar} \]

   9. Taylor expanded in NdChar around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NdChar} \]
   10. Step-by-step derivation

   11. Applied rewrites 27.2%

       \[\leadsto 0.5 \cdot \color{blue}{NdChar} \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq -1 \cdot 10^{-214}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 0:\\ \;\;\;\;\frac{1}{\frac{1 + \frac{\frac{NdChar \cdot NdChar}{NaChar} - NdChar}{NaChar}}{NaChar}} \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 5 \cdot 10^{-103}:\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{else}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 33.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NaChar + NdChar\right) \cdot 0.5\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-214}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(-NaChar, NdChar, NdChar \cdot NdChar\right)}{NaChar \cdot NaChar}}{NaChar}} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-103}:\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ NaChar NdChar) 0.5))
        (t_1
         (-
          (/ NdChar (+ 1.0 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT))))
          (/ NaChar (- -1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))))))
   (if (<= t_1 -1e-214)
     t_0
     (if (<= t_1 0.0)
       (*
        (/
         1.0
         (/
          (/ (fma (- NaChar) NdChar (* NdChar NdChar)) (* NaChar NaChar))
          NaChar))
        0.5)
       (if (<= t_1 5e-103) (* 0.5 NdChar) t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	double tmp;
	if (t_1 <= -1e-214) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (1.0 / ((fma(-NaChar, NdChar, (NdChar * NdChar)) / (NaChar * NaChar)) / NaChar)) * 0.5;
	} else if (t_1 <= 5e-103) {
		tmp = 0.5 * NdChar;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar + NdChar) * 0.5)
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)))) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)))))
	tmp = 0.0
	if (t_1 <= -1e-214)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(1.0 / Float64(Float64(fma(Float64(-NaChar), NdChar, Float64(NdChar * NdChar)) / Float64(NaChar * NaChar)) / NaChar)) * 0.5);
	elseif (t_1 <= 5e-103)
		tmp = Float64(0.5 * NdChar);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar + NdChar), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-214], t$95$0, If[LessEqual[t$95$1, 0.0], N[(N[(1.0 / N[(N[(N[((-NaChar) * NdChar + N[(NdChar * NdChar), $MachinePrecision]), $MachinePrecision] / N[(NaChar * NaChar), $MachinePrecision]), $MachinePrecision] / NaChar), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 5e-103], N[(0.5 * NdChar), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -9.99999999999999913e-215 or 4.99999999999999966e-103 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 38.7

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 38.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

   if -9.99999999999999913e-215 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 2.5

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 2.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 3.9%

      \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{NdChar - NaChar}{\left(NaChar + NdChar\right) \cdot \left(NdChar - NaChar\right)}}} \]

   8. Taylor expanded in NaChar around inf

      \[\leadsto \frac{1}{2} \cdot \frac{1}{\frac{\left(1 + \frac{{NdChar}^{2}}{{NaChar}^{2}}\right) - \frac{NdChar}{NaChar}}{\color{blue}{NaChar}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 23.1%

       \[\leadsto 0.5 \cdot \frac{1}{\frac{\mathsf{fma}\left(\frac{NdChar}{NaChar}, \frac{NdChar}{NaChar}, 1\right) - \frac{NdChar}{NaChar}}{\color{blue}{NaChar}}} \]

   11. Taylor expanded in NaChar around 0

       \[\leadsto \frac{1}{2} \cdot \frac{1}{\frac{\frac{-1 \cdot \left(NaChar \cdot NdChar\right) + {NdChar}^{2}}{{NaChar}^{2}}}{NaChar}} \]
   12. Step-by-step derivation

   13. Applied rewrites 34.0%

       \[\leadsto 0.5 \cdot \frac{1}{\frac{\frac{\mathsf{fma}\left(-NaChar, NdChar, NdChar \cdot NdChar\right)}{NaChar \cdot NaChar}}{NaChar}} \]

   if 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999966e-103

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 9.1

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 9.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

   6. Taylor expanded in NdChar around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NaChar} \]
   7. Step-by-step derivation

   8. Applied rewrites 13.0%

      \[\leadsto 0.5 \cdot \color{blue}{NaChar} \]

   9. Taylor expanded in NdChar around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NdChar} \]
   10. Step-by-step derivation

   11. Applied rewrites 27.2%

       \[\leadsto 0.5 \cdot \color{blue}{NdChar} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq -1 \cdot 10^{-214}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 0:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(-NaChar, NdChar, NdChar \cdot NdChar\right)}{NaChar \cdot NaChar}}{NaChar}} \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 5 \cdot 10^{-103}:\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{else}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 33.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NaChar + NdChar\right) \cdot 0.5\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-214}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{1}{\frac{\frac{NdChar}{NaChar} \cdot \frac{NdChar}{NaChar}}{NaChar}} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-103}:\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ NaChar NdChar) 0.5))
        (t_1
         (-
          (/ NdChar (+ 1.0 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT))))
          (/ NaChar (- -1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))))))
   (if (<= t_1 -1e-214)
     t_0
     (if (<= t_1 0.0)
       (* (/ 1.0 (/ (* (/ NdChar NaChar) (/ NdChar NaChar)) NaChar)) 0.5)
       (if (<= t_1 5e-103) (* 0.5 NdChar) t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	double tmp;
	if (t_1 <= -1e-214) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (1.0 / (((NdChar / NaChar) * (NdChar / NaChar)) / NaChar)) * 0.5;
	} else if (t_1 <= 5e-103) {
		tmp = 0.5 * NdChar;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (nachar + ndchar) * 0.5d0
    t_1 = (ndchar / (1.0d0 + exp(((mu - ((ec - vef) - edonor)) / kbt)))) - (nachar / ((-1.0d0) - exp((((eaccept + (ev + vef)) - mu) / kbt))))
    if (t_1 <= (-1d-214)) then
        tmp = t_0
    else if (t_1 <= 0.0d0) then
        tmp = (1.0d0 / (((ndchar / nachar) * (ndchar / nachar)) / nachar)) * 0.5d0
    else if (t_1 <= 5d-103) then
        tmp = 0.5d0 * ndchar
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (1.0 + Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	double tmp;
	if (t_1 <= -1e-214) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (1.0 / (((NdChar / NaChar) * (NdChar / NaChar)) / NaChar)) * 0.5;
	} else if (t_1 <= 5e-103) {
		tmp = 0.5 * NdChar;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar + NdChar) * 0.5
	t_1 = (NdChar / (1.0 + math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))))
	tmp = 0
	if t_1 <= -1e-214:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = (1.0 / (((NdChar / NaChar) * (NdChar / NaChar)) / NaChar)) * 0.5
	elif t_1 <= 5e-103:
		tmp = 0.5 * NdChar
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar + NdChar) * 0.5)
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)))) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)))))
	tmp = 0.0
	if (t_1 <= -1e-214)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(NdChar / NaChar) * Float64(NdChar / NaChar)) / NaChar)) * 0.5);
	elseif (t_1 <= 5e-103)
		tmp = Float64(0.5 * NdChar);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NaChar + NdChar) * 0.5;
	t_1 = (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	tmp = 0.0;
	if (t_1 <= -1e-214)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = (1.0 / (((NdChar / NaChar) * (NdChar / NaChar)) / NaChar)) * 0.5;
	elseif (t_1 <= 5e-103)
		tmp = 0.5 * NdChar;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar + NdChar), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-214], t$95$0, If[LessEqual[t$95$1, 0.0], N[(N[(1.0 / N[(N[(N[(NdChar / NaChar), $MachinePrecision] * N[(NdChar / NaChar), $MachinePrecision]), $MachinePrecision] / NaChar), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 5e-103], N[(0.5 * NdChar), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -9.99999999999999913e-215 or 4.99999999999999966e-103 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 38.7

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 38.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

   if -9.99999999999999913e-215 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 2.5

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 2.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 3.9%

      \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{NdChar - NaChar}{\left(NaChar + NdChar\right) \cdot \left(NdChar - NaChar\right)}}} \]

   8. Taylor expanded in NaChar around inf

      \[\leadsto \frac{1}{2} \cdot \frac{1}{\frac{\left(1 + \frac{{NdChar}^{2}}{{NaChar}^{2}}\right) - \frac{NdChar}{NaChar}}{\color{blue}{NaChar}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 23.1%

       \[\leadsto 0.5 \cdot \frac{1}{\frac{\mathsf{fma}\left(\frac{NdChar}{NaChar}, \frac{NdChar}{NaChar}, 1\right) - \frac{NdChar}{NaChar}}{\color{blue}{NaChar}}} \]

   11. Taylor expanded in NdChar around inf

       \[\leadsto \frac{1}{2} \cdot \frac{1}{\frac{\frac{{NdChar}^{2}}{{NaChar}^{2}}}{NaChar}} \]
   12. Step-by-step derivation

   13. Applied rewrites 32.7%

       \[\leadsto 0.5 \cdot \frac{1}{\frac{\frac{NdChar}{NaChar} \cdot \frac{NdChar}{NaChar}}{NaChar}} \]

   if 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999966e-103

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 9.1

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 9.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

   6. Taylor expanded in NdChar around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NaChar} \]
   7. Step-by-step derivation

   8. Applied rewrites 13.0%

      \[\leadsto 0.5 \cdot \color{blue}{NaChar} \]

   9. Taylor expanded in NdChar around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NdChar} \]
   10. Step-by-step derivation

   11. Applied rewrites 27.2%

       \[\leadsto 0.5 \cdot \color{blue}{NdChar} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq -1 \cdot 10^{-214}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 0:\\ \;\;\;\;\frac{1}{\frac{\frac{NdChar}{NaChar} \cdot \frac{NdChar}{NaChar}}{NaChar}} \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 5 \cdot 10^{-103}:\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{else}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 32.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NaChar + NdChar\right) \cdot 0.5\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-214}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{1}{\frac{1}{NaChar} - \frac{NdChar}{NaChar \cdot NaChar}} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-103}:\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ NaChar NdChar) 0.5))
        (t_1
         (-
          (/ NdChar (+ 1.0 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT))))
          (/ NaChar (- -1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))))))
   (if (<= t_1 -1e-214)
     t_0
     (if (<= t_1 0.0)
       (* (/ 1.0 (- (/ 1.0 NaChar) (/ NdChar (* NaChar NaChar)))) 0.5)
       (if (<= t_1 5e-103) (* 0.5 NdChar) t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	double tmp;
	if (t_1 <= -1e-214) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (1.0 / ((1.0 / NaChar) - (NdChar / (NaChar * NaChar)))) * 0.5;
	} else if (t_1 <= 5e-103) {
		tmp = 0.5 * NdChar;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (nachar + ndchar) * 0.5d0
    t_1 = (ndchar / (1.0d0 + exp(((mu - ((ec - vef) - edonor)) / kbt)))) - (nachar / ((-1.0d0) - exp((((eaccept + (ev + vef)) - mu) / kbt))))
    if (t_1 <= (-1d-214)) then
        tmp = t_0
    else if (t_1 <= 0.0d0) then
        tmp = (1.0d0 / ((1.0d0 / nachar) - (ndchar / (nachar * nachar)))) * 0.5d0
    else if (t_1 <= 5d-103) then
        tmp = 0.5d0 * ndchar
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (1.0 + Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	double tmp;
	if (t_1 <= -1e-214) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (1.0 / ((1.0 / NaChar) - (NdChar / (NaChar * NaChar)))) * 0.5;
	} else if (t_1 <= 5e-103) {
		tmp = 0.5 * NdChar;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar + NdChar) * 0.5
	t_1 = (NdChar / (1.0 + math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))))
	tmp = 0
	if t_1 <= -1e-214:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = (1.0 / ((1.0 / NaChar) - (NdChar / (NaChar * NaChar)))) * 0.5
	elif t_1 <= 5e-103:
		tmp = 0.5 * NdChar
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar + NdChar) * 0.5)
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)))) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)))))
	tmp = 0.0
	if (t_1 <= -1e-214)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(1.0 / Float64(Float64(1.0 / NaChar) - Float64(NdChar / Float64(NaChar * NaChar)))) * 0.5);
	elseif (t_1 <= 5e-103)
		tmp = Float64(0.5 * NdChar);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NaChar + NdChar) * 0.5;
	t_1 = (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	tmp = 0.0;
	if (t_1 <= -1e-214)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = (1.0 / ((1.0 / NaChar) - (NdChar / (NaChar * NaChar)))) * 0.5;
	elseif (t_1 <= 5e-103)
		tmp = 0.5 * NdChar;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar + NdChar), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-214], t$95$0, If[LessEqual[t$95$1, 0.0], N[(N[(1.0 / N[(N[(1.0 / NaChar), $MachinePrecision] - N[(NdChar / N[(NaChar * NaChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 5e-103], N[(0.5 * NdChar), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -9.99999999999999913e-215 or 4.99999999999999966e-103 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 38.7

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 38.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

   if -9.99999999999999913e-215 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 2.5

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 2.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 3.9%

      \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{NdChar - NaChar}{\left(NaChar + NdChar\right) \cdot \left(NdChar - NaChar\right)}}} \]

   8. Taylor expanded in NdChar around 0

      \[\leadsto \frac{1}{2} \cdot \frac{1}{-1 \cdot \frac{NdChar}{{NaChar}^{2}} + \color{blue}{\frac{1}{NaChar}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 28.5%

       \[\leadsto 0.5 \cdot \frac{1}{\frac{1}{NaChar} - \color{blue}{\frac{NdChar}{NaChar \cdot NaChar}}} \]

   if 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999966e-103

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 9.1

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 9.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

   6. Taylor expanded in NdChar around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NaChar} \]
   7. Step-by-step derivation

   8. Applied rewrites 13.0%

      \[\leadsto 0.5 \cdot \color{blue}{NaChar} \]

   9. Taylor expanded in NdChar around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NdChar} \]
   10. Step-by-step derivation

   11. Applied rewrites 27.2%

       \[\leadsto 0.5 \cdot \color{blue}{NdChar} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq -1 \cdot 10^{-214}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 0:\\ \;\;\;\;\frac{1}{\frac{1}{NaChar} - \frac{NdChar}{NaChar \cdot NaChar}} \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 5 \cdot 10^{-103}:\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{else}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 32.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NaChar + NdChar\right) \cdot 0.5\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-214}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{1}{\frac{1 - \frac{NdChar}{NaChar}}{NaChar}} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-103}:\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ NaChar NdChar) 0.5))
        (t_1
         (-
          (/ NdChar (+ 1.0 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT))))
          (/ NaChar (- -1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))))))
   (if (<= t_1 -1e-214)
     t_0
     (if (<= t_1 0.0)
       (* (/ 1.0 (/ (- 1.0 (/ NdChar NaChar)) NaChar)) 0.5)
       (if (<= t_1 5e-103) (* 0.5 NdChar) t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	double tmp;
	if (t_1 <= -1e-214) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (1.0 / ((1.0 - (NdChar / NaChar)) / NaChar)) * 0.5;
	} else if (t_1 <= 5e-103) {
		tmp = 0.5 * NdChar;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (nachar + ndchar) * 0.5d0
    t_1 = (ndchar / (1.0d0 + exp(((mu - ((ec - vef) - edonor)) / kbt)))) - (nachar / ((-1.0d0) - exp((((eaccept + (ev + vef)) - mu) / kbt))))
    if (t_1 <= (-1d-214)) then
        tmp = t_0
    else if (t_1 <= 0.0d0) then
        tmp = (1.0d0 / ((1.0d0 - (ndchar / nachar)) / nachar)) * 0.5d0
    else if (t_1 <= 5d-103) then
        tmp = 0.5d0 * ndchar
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (1.0 + Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	double tmp;
	if (t_1 <= -1e-214) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (1.0 / ((1.0 - (NdChar / NaChar)) / NaChar)) * 0.5;
	} else if (t_1 <= 5e-103) {
		tmp = 0.5 * NdChar;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar + NdChar) * 0.5
	t_1 = (NdChar / (1.0 + math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))))
	tmp = 0
	if t_1 <= -1e-214:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = (1.0 / ((1.0 - (NdChar / NaChar)) / NaChar)) * 0.5
	elif t_1 <= 5e-103:
		tmp = 0.5 * NdChar
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar + NdChar) * 0.5)
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)))) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)))))
	tmp = 0.0
	if (t_1 <= -1e-214)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(1.0 / Float64(Float64(1.0 - Float64(NdChar / NaChar)) / NaChar)) * 0.5);
	elseif (t_1 <= 5e-103)
		tmp = Float64(0.5 * NdChar);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NaChar + NdChar) * 0.5;
	t_1 = (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	tmp = 0.0;
	if (t_1 <= -1e-214)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = (1.0 / ((1.0 - (NdChar / NaChar)) / NaChar)) * 0.5;
	elseif (t_1 <= 5e-103)
		tmp = 0.5 * NdChar;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar + NdChar), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-214], t$95$0, If[LessEqual[t$95$1, 0.0], N[(N[(1.0 / N[(N[(1.0 - N[(NdChar / NaChar), $MachinePrecision]), $MachinePrecision] / NaChar), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 5e-103], N[(0.5 * NdChar), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -9.99999999999999913e-215 or 4.99999999999999966e-103 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 38.7

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 38.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

   if -9.99999999999999913e-215 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 2.5

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 2.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 3.9%

      \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{NdChar - NaChar}{\left(NaChar + NdChar\right) \cdot \left(NdChar - NaChar\right)}}} \]

   8. Taylor expanded in NaChar around inf

      \[\leadsto \frac{1}{2} \cdot \frac{1}{\frac{1 + -1 \cdot \frac{NdChar}{NaChar}}{\color{blue}{NaChar}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 28.4%

       \[\leadsto 0.5 \cdot \frac{1}{\frac{1 - \frac{NdChar}{NaChar}}{\color{blue}{NaChar}}} \]

   if 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999966e-103

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 9.1

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 9.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

   6. Taylor expanded in NdChar around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NaChar} \]
   7. Step-by-step derivation

   8. Applied rewrites 13.0%

      \[\leadsto 0.5 \cdot \color{blue}{NaChar} \]

   9. Taylor expanded in NdChar around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NdChar} \]
   10. Step-by-step derivation

   11. Applied rewrites 27.2%

       \[\leadsto 0.5 \cdot \color{blue}{NdChar} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq -1 \cdot 10^{-214}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 0:\\ \;\;\;\;\frac{1}{\frac{1 - \frac{NdChar}{NaChar}}{NaChar}} \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 5 \cdot 10^{-103}:\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{else}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NaChar + NdChar\right) \cdot 0.5\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-228}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{1}{\frac{NdChar - NaChar}{NdChar \cdot NdChar}} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-103}:\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ NaChar NdChar) 0.5))
        (t_1
         (-
          (/ NdChar (+ 1.0 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT))))
          (/ NaChar (- -1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))))))
   (if (<= t_1 -4e-228)
     t_0
     (if (<= t_1 0.0)
       (* (/ 1.0 (/ (- NdChar NaChar) (* NdChar NdChar))) 0.5)
       (if (<= t_1 5e-103) (* 0.5 NdChar) t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	double tmp;
	if (t_1 <= -4e-228) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (1.0 / ((NdChar - NaChar) / (NdChar * NdChar))) * 0.5;
	} else if (t_1 <= 5e-103) {
		tmp = 0.5 * NdChar;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (nachar + ndchar) * 0.5d0
    t_1 = (ndchar / (1.0d0 + exp(((mu - ((ec - vef) - edonor)) / kbt)))) - (nachar / ((-1.0d0) - exp((((eaccept + (ev + vef)) - mu) / kbt))))
    if (t_1 <= (-4d-228)) then
        tmp = t_0
    else if (t_1 <= 0.0d0) then
        tmp = (1.0d0 / ((ndchar - nachar) / (ndchar * ndchar))) * 0.5d0
    else if (t_1 <= 5d-103) then
        tmp = 0.5d0 * ndchar
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (1.0 + Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	double tmp;
	if (t_1 <= -4e-228) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (1.0 / ((NdChar - NaChar) / (NdChar * NdChar))) * 0.5;
	} else if (t_1 <= 5e-103) {
		tmp = 0.5 * NdChar;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar + NdChar) * 0.5
	t_1 = (NdChar / (1.0 + math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))))
	tmp = 0
	if t_1 <= -4e-228:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = (1.0 / ((NdChar - NaChar) / (NdChar * NdChar))) * 0.5
	elif t_1 <= 5e-103:
		tmp = 0.5 * NdChar
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar + NdChar) * 0.5)
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)))) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)))))
	tmp = 0.0
	if (t_1 <= -4e-228)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(1.0 / Float64(Float64(NdChar - NaChar) / Float64(NdChar * NdChar))) * 0.5);
	elseif (t_1 <= 5e-103)
		tmp = Float64(0.5 * NdChar);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NaChar + NdChar) * 0.5;
	t_1 = (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	tmp = 0.0;
	if (t_1 <= -4e-228)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = (1.0 / ((NdChar - NaChar) / (NdChar * NdChar))) * 0.5;
	elseif (t_1 <= 5e-103)
		tmp = 0.5 * NdChar;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar + NdChar), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-228], t$95$0, If[LessEqual[t$95$1, 0.0], N[(N[(1.0 / N[(N[(NdChar - NaChar), $MachinePrecision] / N[(NdChar * NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 5e-103], N[(0.5 * NdChar), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4.00000000000000013e-228 or 4.99999999999999966e-103 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 38.0

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 38.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

   if -4.00000000000000013e-228 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 2.6

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 2.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 4.0%

      \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{NdChar - NaChar}{\left(NaChar + NdChar\right) \cdot \left(NdChar - NaChar\right)}}} \]

   8. Taylor expanded in NdChar around inf

      \[\leadsto \frac{1}{2} \cdot \frac{1}{\frac{1 + -1 \cdot \frac{NaChar}{NdChar}}{\color{blue}{NdChar}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 19.9%

       \[\leadsto 0.5 \cdot \frac{1}{\frac{1 - \frac{NaChar}{NdChar}}{\color{blue}{NdChar}}} \]

   11. Taylor expanded in NdChar around 0

       \[\leadsto \frac{1}{2} \cdot \frac{1}{\frac{NdChar + -1 \cdot NaChar}{{NdChar}^{\color{blue}{2}}}} \]
   12. Step-by-step derivation

   13. Applied rewrites 22.9%

       \[\leadsto 0.5 \cdot \frac{1}{\frac{NdChar - NaChar}{NdChar \cdot \color{blue}{NdChar}}} \]

   if 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999966e-103

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 9.1

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 9.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

   6. Taylor expanded in NdChar around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NaChar} \]
   7. Step-by-step derivation

   8. Applied rewrites 13.0%

      \[\leadsto 0.5 \cdot \color{blue}{NaChar} \]

   9. Taylor expanded in NdChar around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NdChar} \]
   10. Step-by-step derivation

   11. Applied rewrites 27.2%

       \[\leadsto 0.5 \cdot \color{blue}{NdChar} \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq -4 \cdot 10^{-228}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 0:\\ \;\;\;\;\frac{1}{\frac{NdChar - NaChar}{NdChar \cdot NdChar}} \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 5 \cdot 10^{-103}:\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{else}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 32.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NaChar + NdChar\right) \cdot 0.5\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-228}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{1}{\frac{-NaChar}{NdChar \cdot NdChar}} \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-103}:\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ NaChar NdChar) 0.5))
        (t_1
         (-
          (/ NdChar (+ 1.0 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT))))
          (/ NaChar (- -1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))))))
   (if (<= t_1 -4e-228)
     t_0
     (if (<= t_1 0.0)
       (* (/ 1.0 (/ (- NaChar) (* NdChar NdChar))) 0.5)
       (if (<= t_1 5e-103) (* 0.5 NdChar) t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	double tmp;
	if (t_1 <= -4e-228) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (1.0 / (-NaChar / (NdChar * NdChar))) * 0.5;
	} else if (t_1 <= 5e-103) {
		tmp = 0.5 * NdChar;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (nachar + ndchar) * 0.5d0
    t_1 = (ndchar / (1.0d0 + exp(((mu - ((ec - vef) - edonor)) / kbt)))) - (nachar / ((-1.0d0) - exp((((eaccept + (ev + vef)) - mu) / kbt))))
    if (t_1 <= (-4d-228)) then
        tmp = t_0
    else if (t_1 <= 0.0d0) then
        tmp = (1.0d0 / (-nachar / (ndchar * ndchar))) * 0.5d0
    else if (t_1 <= 5d-103) then
        tmp = 0.5d0 * ndchar
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (1.0 + Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	double tmp;
	if (t_1 <= -4e-228) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (1.0 / (-NaChar / (NdChar * NdChar))) * 0.5;
	} else if (t_1 <= 5e-103) {
		tmp = 0.5 * NdChar;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar + NdChar) * 0.5
	t_1 = (NdChar / (1.0 + math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))))
	tmp = 0
	if t_1 <= -4e-228:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = (1.0 / (-NaChar / (NdChar * NdChar))) * 0.5
	elif t_1 <= 5e-103:
		tmp = 0.5 * NdChar
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar + NdChar) * 0.5)
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)))) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)))))
	tmp = 0.0
	if (t_1 <= -4e-228)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(1.0 / Float64(Float64(-NaChar) / Float64(NdChar * NdChar))) * 0.5);
	elseif (t_1 <= 5e-103)
		tmp = Float64(0.5 * NdChar);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NaChar + NdChar) * 0.5;
	t_1 = (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	tmp = 0.0;
	if (t_1 <= -4e-228)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = (1.0 / (-NaChar / (NdChar * NdChar))) * 0.5;
	elseif (t_1 <= 5e-103)
		tmp = 0.5 * NdChar;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar + NdChar), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-228], t$95$0, If[LessEqual[t$95$1, 0.0], N[(N[(1.0 / N[((-NaChar) / N[(NdChar * NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 5e-103], N[(0.5 * NdChar), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4.00000000000000013e-228 or 4.99999999999999966e-103 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 38.0

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 38.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

   if -4.00000000000000013e-228 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 2.6

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 2.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 4.0%

      \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{NdChar - NaChar}{\left(NaChar + NdChar\right) \cdot \left(NdChar - NaChar\right)}}} \]

   8. Taylor expanded in NdChar around inf

      \[\leadsto \frac{1}{2} \cdot \frac{1}{\frac{1 + -1 \cdot \frac{NaChar}{NdChar}}{\color{blue}{NdChar}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 19.9%

       \[\leadsto 0.5 \cdot \frac{1}{\frac{1 - \frac{NaChar}{NdChar}}{\color{blue}{NdChar}}} \]

   11. Taylor expanded in NdChar around 0

       \[\leadsto \frac{1}{2} \cdot \frac{1}{-1 \cdot \frac{NaChar}{\color{blue}{{NdChar}^{2}}}} \]
   12. Step-by-step derivation

   13. Applied rewrites 22.8%

       \[\leadsto 0.5 \cdot \frac{1}{\frac{-NaChar}{NdChar \cdot \color{blue}{NdChar}}} \]

   if 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999966e-103

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 9.1

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 9.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

   6. Taylor expanded in NdChar around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NaChar} \]
   7. Step-by-step derivation

   8. Applied rewrites 13.0%

      \[\leadsto 0.5 \cdot \color{blue}{NaChar} \]

   9. Taylor expanded in NdChar around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NdChar} \]
   10. Step-by-step derivation

   11. Applied rewrites 27.2%

       \[\leadsto 0.5 \cdot \color{blue}{NdChar} \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq -4 \cdot 10^{-228}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 0:\\ \;\;\;\;\frac{1}{\frac{-NaChar}{NdChar \cdot NdChar}} \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 5 \cdot 10^{-103}:\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{else}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 29.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(NaChar + NdChar\right) \cdot 0.5\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-218}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(\frac{NaChar}{KbT} \cdot EAccept\right) \cdot -0.25\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-103}:\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (+ NaChar NdChar) 0.5))
        (t_1
         (-
          (/ NdChar (+ 1.0 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT))))
          (/ NaChar (- -1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))))))
   (if (<= t_1 -5e-218)
     t_0
     (if (<= t_1 0.0)
       (* (* (/ NaChar KbT) EAccept) -0.25)
       (if (<= t_1 5e-103) (* 0.5 NdChar) t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	double tmp;
	if (t_1 <= -5e-218) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = ((NaChar / KbT) * EAccept) * -0.25;
	} else if (t_1 <= 5e-103) {
		tmp = 0.5 * NdChar;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (nachar + ndchar) * 0.5d0
    t_1 = (ndchar / (1.0d0 + exp(((mu - ((ec - vef) - edonor)) / kbt)))) - (nachar / ((-1.0d0) - exp((((eaccept + (ev + vef)) - mu) / kbt))))
    if (t_1 <= (-5d-218)) then
        tmp = t_0
    else if (t_1 <= 0.0d0) then
        tmp = ((nachar / kbt) * eaccept) * (-0.25d0)
    else if (t_1 <= 5d-103) then
        tmp = 0.5d0 * ndchar
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar + NdChar) * 0.5;
	double t_1 = (NdChar / (1.0 + Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	double tmp;
	if (t_1 <= -5e-218) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = ((NaChar / KbT) * EAccept) * -0.25;
	} else if (t_1 <= 5e-103) {
		tmp = 0.5 * NdChar;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar + NdChar) * 0.5
	t_1 = (NdChar / (1.0 + math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))))
	tmp = 0
	if t_1 <= -5e-218:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = ((NaChar / KbT) * EAccept) * -0.25
	elif t_1 <= 5e-103:
		tmp = 0.5 * NdChar
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar + NdChar) * 0.5)
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)))) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)))))
	tmp = 0.0
	if (t_1 <= -5e-218)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(NaChar / KbT) * EAccept) * -0.25);
	elseif (t_1 <= 5e-103)
		tmp = Float64(0.5 * NdChar);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NaChar + NdChar) * 0.5;
	t_1 = (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	tmp = 0.0;
	if (t_1 <= -5e-218)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = ((NaChar / KbT) * EAccept) * -0.25;
	elseif (t_1 <= 5e-103)
		tmp = 0.5 * NdChar;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar + NdChar), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-218], t$95$0, If[LessEqual[t$95$1, 0.0], N[(N[(N[(NaChar / KbT), $MachinePrecision] * EAccept), $MachinePrecision] * -0.25), $MachinePrecision], If[LessEqual[t$95$1, 5e-103], N[(0.5 * NdChar), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -5.00000000000000041e-218 or 4.99999999999999966e-103 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 38.5

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 38.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

   if -5.00000000000000041e-218 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right) + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(\frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right)} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}, \frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)} \]

   5. Applied rewrites 1.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, \mathsf{fma}\left(NaChar, \frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}, NdChar \cdot \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}\right), 0.5 \cdot \left(NdChar + NaChar\right)\right)} \]

   6. Taylor expanded in EAccept around inf

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{EAccept \cdot NaChar}{KbT}} \]
   7. Step-by-step derivation

   8. Applied rewrites 14.4%

      \[\leadsto \left(EAccept \cdot \frac{NaChar}{KbT}\right) \cdot \color{blue}{-0.25} \]

   if 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.99999999999999966e-103

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 9.1

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 9.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

   6. Taylor expanded in NdChar around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NaChar} \]
   7. Step-by-step derivation

   8. Applied rewrites 13.0%

      \[\leadsto 0.5 \cdot \color{blue}{NaChar} \]

   9. Taylor expanded in NdChar around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NdChar} \]
   10. Step-by-step derivation

   11. Applied rewrites 27.2%

       \[\leadsto 0.5 \cdot \color{blue}{NdChar} \]
3. Recombined 3 regimes into one program.

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq -5 \cdot 10^{-218}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 0:\\ \;\;\;\;\left(\frac{NaChar}{KbT} \cdot EAccept\right) \cdot -0.25\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 5 \cdot 10^{-103}:\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{else}:\\ \;\;\;\;\left(NaChar + NdChar\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}\\ t_1 := \frac{NdChar}{1 + t\_0} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ t_2 := \frac{NdChar}{-1 - t\_0}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-250}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} - t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-241}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} - t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT)))
        (t_1
         (-
          (/ NdChar (+ 1.0 t_0))
          (/ NaChar (- -1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT))))))
        (t_2 (/ NdChar (- -1.0 t_0))))
   (if (<= t_1 -2e-250)
     (- (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) t_2)
     (if (<= t_1 5e-241)
       (/ NdChar (+ 1.0 (exp (/ (- (+ (+ mu Vef) EDonor) Ec) KbT))))
       (- (/ NaChar (+ 1.0 (exp (/ Ev KbT)))) t_2)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp(((mu - ((Ec - Vef) - EDonor)) / KbT));
	double t_1 = (NdChar / (1.0 + t_0)) - (NaChar / (-1.0 - exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	double t_2 = NdChar / (-1.0 - t_0);
	double tmp;
	if (t_1 <= -2e-250) {
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) - t_2;
	} else if (t_1 <= 5e-241) {
		tmp = NdChar / (1.0 + exp(((((mu + Vef) + EDonor) - Ec) / KbT)));
	} else {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = exp(((mu - ((ec - vef) - edonor)) / kbt))
    t_1 = (ndchar / (1.0d0 + t_0)) - (nachar / ((-1.0d0) - exp((((eaccept + (ev + vef)) - mu) / kbt))))
    t_2 = ndchar / ((-1.0d0) - t_0)
    if (t_1 <= (-2d-250)) then
        tmp = (nachar / (1.0d0 + exp((eaccept / kbt)))) - t_2
    else if (t_1 <= 5d-241) then
        tmp = ndchar / (1.0d0 + exp(((((mu + vef) + edonor) - ec) / kbt)))
    else
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) - t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT));
	double t_1 = (NdChar / (1.0 + t_0)) - (NaChar / (-1.0 - Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	double t_2 = NdChar / (-1.0 - t_0);
	double tmp;
	if (t_1 <= -2e-250) {
		tmp = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) - t_2;
	} else if (t_1 <= 5e-241) {
		tmp = NdChar / (1.0 + Math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)));
	} else {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) - t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT))
	t_1 = (NdChar / (1.0 + t_0)) - (NaChar / (-1.0 - math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))))
	t_2 = NdChar / (-1.0 - t_0)
	tmp = 0
	if t_1 <= -2e-250:
		tmp = (NaChar / (1.0 + math.exp((EAccept / KbT)))) - t_2
	elif t_1 <= 5e-241:
		tmp = NdChar / (1.0 + math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)))
	else:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) - t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + t_0)) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)))))
	t_2 = Float64(NdChar / Float64(-1.0 - t_0))
	tmp = 0.0
	if (t_1 <= -2e-250)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) - t_2);
	elseif (t_1 <= 5e-241)
		tmp = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT))));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((mu - ((Ec - Vef) - EDonor)) / KbT));
	t_1 = (NdChar / (1.0 + t_0)) - (NaChar / (-1.0 - exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	t_2 = NdChar / (-1.0 - t_0);
	tmp = 0.0;
	if (t_1 <= -2e-250)
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) - t_2;
	elseif (t_1 <= 5e-241)
		tmp = NdChar / (1.0 + exp(((((mu + Vef) + EDonor) - Ec) / KbT)));
	else
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(NdChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-250], N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 5e-241], N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2.0000000000000001e-250

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in EAccept around inf

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 81.4

         \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]

   5. Applied rewrites 81.4%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]

   if -2.0000000000000001e-250 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.9999999999999998e-241

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 3.2

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 3.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

   6. Taylor expanded in NdChar around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NaChar} \]
   7. Step-by-step derivation

   8. Applied rewrites 4.7%

      \[\leadsto 0.5 \cdot \color{blue}{NaChar} \]

   9. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

       2. +-commutative N/A

          \[\leadsto \frac{NdChar}{\color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}} + 1}} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{NdChar}{\color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}} + 1}} \]

       4. lower-exp.f64 N/A

          \[\leadsto \frac{NdChar}{\color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}} + 1} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{NdChar}{e^{\color{blue}{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}} + 1} \]

       6. lower--.f64 N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\color{blue}{\left(EDonor + \left(Vef + mu\right)\right) - Ec}}{KbT}} + 1} \]

       7. +-commutative N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\color{blue}{\left(\left(Vef + mu\right) + EDonor\right)} - Ec}{KbT}} + 1} \]

       8. lower-+.f64 N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\color{blue}{\left(\left(Vef + mu\right) + EDonor\right)} - Ec}{KbT}} + 1} \]

       9. +-commutative N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\left(\color{blue}{\left(mu + Vef\right)} + EDonor\right) - Ec}{KbT}} + 1} \]

       10. lower-+.f64 95.5

           \[\leadsto \frac{NdChar}{e^{\frac{\left(\color{blue}{\left(mu + Vef\right)} + EDonor\right) - Ec}{KbT}} + 1} \]

   11. Applied rewrites 95.5%

       \[\leadsto \color{blue}{\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + 1}} \]

   if 4.9999999999999998e-241 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in Ev around inf

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{Ev}{KbT}}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 71.5

         \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{Ev}{KbT}}}} \]

   5. Applied rewrites 71.5%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{Ev}{KbT}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq -2 \cdot 10^{-250}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 5 \cdot 10^{-241}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(mu + Vef\right) + EDonor\\ t_1 := e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\\ t_2 := \frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - t\_1}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-228}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(mu + Vef\right) - Ec}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{t\_0 - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{\mathsf{fma}\left(\frac{Ec - t\_0}{KbT}, -1, 2\right)} + \frac{NaChar}{1 + t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (+ mu Vef) EDonor))
        (t_1 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))
        (t_2
         (-
          (/ NdChar (+ 1.0 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT))))
          (/ NaChar (- -1.0 t_1)))))
   (if (<= t_2 -4e-228)
     (-
      (/ NdChar (+ 1.0 (exp (/ (- (+ mu Vef) Ec) KbT))))
      (/ NaChar (- -1.0 (exp (/ Vef KbT)))))
     (if (<= t_2 5e-70)
       (/ NdChar (+ 1.0 (exp (/ (- t_0 Ec) KbT))))
       (+
        (/ NdChar (fma (/ (- Ec t_0) KbT) -1.0 2.0))
        (/ NaChar (+ 1.0 t_1)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (mu + Vef) + EDonor;
	double t_1 = exp((((EAccept + (Ev + Vef)) - mu) / KbT));
	double t_2 = (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - t_1));
	double tmp;
	if (t_2 <= -4e-228) {
		tmp = (NdChar / (1.0 + exp((((mu + Vef) - Ec) / KbT)))) - (NaChar / (-1.0 - exp((Vef / KbT))));
	} else if (t_2 <= 5e-70) {
		tmp = NdChar / (1.0 + exp(((t_0 - Ec) / KbT)));
	} else {
		tmp = (NdChar / fma(((Ec - t_0) / KbT), -1.0, 2.0)) + (NaChar / (1.0 + t_1));
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(mu + Vef) + EDonor)
	t_1 = exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))
	t_2 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)))) - Float64(NaChar / Float64(-1.0 - t_1)))
	tmp = 0.0
	if (t_2 <= -4e-228)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(mu + Vef) - Ec) / KbT)))) - Float64(NaChar / Float64(-1.0 - exp(Float64(Vef / KbT)))));
	elseif (t_2 <= 5e-70)
		tmp = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(t_0 - Ec) / KbT))));
	else
		tmp = Float64(Float64(NdChar / fma(Float64(Float64(Ec - t_0) / KbT), -1.0, 2.0)) + Float64(NaChar / Float64(1.0 + t_1)));
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-228], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(mu + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-70], N[(NdChar / N[(1.0 + N[Exp[N[(N[(t$95$0 - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(N[(N[(Ec - t$95$0), $MachinePrecision] / KbT), $MachinePrecision] * -1.0 + 2.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4.00000000000000013e-228

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in Vef around inf

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 86.3

         \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   5. Applied rewrites 86.3%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   6. Taylor expanded in EDonor around 0

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(Vef + mu\right) - Ec}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(Vef + mu\right) - Ec}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(mu + Vef\right)} - Ec}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} \]

      3. lower-+.f64 78.7

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(mu + Vef\right)} - Ec}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} \]

   8. Applied rewrites 78.7%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(mu + Vef\right) - Ec}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} \]

   if -4.00000000000000013e-228 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 4.9999999999999998e-70

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 6.6

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 6.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

   6. Taylor expanded in NdChar around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NaChar} \]
   7. Step-by-step derivation

   8. Applied rewrites 7.5%

      \[\leadsto 0.5 \cdot \color{blue}{NaChar} \]

   9. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

       2. +-commutative N/A

          \[\leadsto \frac{NdChar}{\color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}} + 1}} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{NdChar}{\color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}} + 1}} \]

       4. lower-exp.f64 N/A

          \[\leadsto \frac{NdChar}{\color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}} + 1} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{NdChar}{e^{\color{blue}{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}} + 1} \]

       6. lower--.f64 N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\color{blue}{\left(EDonor + \left(Vef + mu\right)\right) - Ec}}{KbT}} + 1} \]

       7. +-commutative N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\color{blue}{\left(\left(Vef + mu\right) + EDonor\right)} - Ec}{KbT}} + 1} \]

       8. lower-+.f64 N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\color{blue}{\left(\left(Vef + mu\right) + EDonor\right)} - Ec}{KbT}} + 1} \]

       9. +-commutative N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\left(\color{blue}{\left(mu + Vef\right)} + EDonor\right) - Ec}{KbT}} + 1} \]

       10. lower-+.f64 84.2

           \[\leadsto \frac{NdChar}{e^{\frac{\left(\color{blue}{\left(mu + Vef\right)} + EDonor\right) - Ec}{KbT}} + 1} \]

   11. Applied rewrites 84.2%

       \[\leadsto \color{blue}{\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + 1}} \]

   if 4.9999999999999998e-70 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \frac{NdChar}{1 + \color{blue}{e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. distribute-frac-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\mathsf{neg}\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. neg-mul-1 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{-1 \cdot \frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. exp-prod N/A

         \[\leadsto \frac{NdChar}{1 + \color{blue}{{\left(e^{-1}\right)}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{NdChar}{1 + \color{blue}{{\left(e^{-1}\right)}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{NdChar}{1 + {\color{blue}{\left(e^{-1}\right)}}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      9. lower-/.f64 99.9

         \[\leadsto \frac{NdChar}{1 + {\left(e^{-1}\right)}^{\color{blue}{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   4. Applied rewrites 99.9%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{{\left(e^{-1}\right)}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   5. Taylor expanded in KbT around inf

      \[\leadsto \frac{NdChar}{\color{blue}{2 + -1 \cdot \frac{Ec - \left(EDonor + \left(Vef + mu\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{NdChar}{\color{blue}{-1 \cdot \frac{Ec - \left(EDonor + \left(Vef + mu\right)\right)}{KbT} + 2}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{NdChar}{\color{blue}{\frac{Ec - \left(EDonor + \left(Vef + mu\right)\right)}{KbT} \cdot -1} + 2} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{NdChar}{\color{blue}{\mathsf{fma}\left(\frac{Ec - \left(EDonor + \left(Vef + mu\right)\right)}{KbT}, -1, 2\right)}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{NdChar}{\mathsf{fma}\left(\color{blue}{\frac{Ec - \left(EDonor + \left(Vef + mu\right)\right)}{KbT}}, -1, 2\right)} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{NdChar}{\mathsf{fma}\left(\frac{\color{blue}{Ec - \left(EDonor + \left(Vef + mu\right)\right)}}{KbT}, -1, 2\right)} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. +-commutative N/A

         \[\leadsto \frac{NdChar}{\mathsf{fma}\left(\frac{Ec - \color{blue}{\left(\left(Vef + mu\right) + EDonor\right)}}{KbT}, -1, 2\right)} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{NdChar}{\mathsf{fma}\left(\frac{Ec - \color{blue}{\left(\left(Vef + mu\right) + EDonor\right)}}{KbT}, -1, 2\right)} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      8. +-commutative N/A

         \[\leadsto \frac{NdChar}{\mathsf{fma}\left(\frac{Ec - \left(\color{blue}{\left(mu + Vef\right)} + EDonor\right)}{KbT}, -1, 2\right)} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      9. lower-+.f64 63.3

         \[\leadsto \frac{NdChar}{\mathsf{fma}\left(\frac{Ec - \left(\color{blue}{\left(mu + Vef\right)} + EDonor\right)}{KbT}, -1, 2\right)} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   7. Applied rewrites 63.3%

      \[\leadsto \frac{NdChar}{\color{blue}{\mathsf{fma}\left(\frac{Ec - \left(\left(mu + Vef\right) + EDonor\right)}{KbT}, -1, 2\right)}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq -4 \cdot 10^{-228}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(mu + Vef\right) - Ec}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;\frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{\mathsf{fma}\left(\frac{Ec - \left(\left(mu + Vef\right) + EDonor\right)}{KbT}, -1, 2\right)} + \frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 92.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ \mathbf{if}\;Vef \leq -4.5 \cdot 10^{+123}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;Vef \leq 1.86 \cdot 10^{+71}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (-
          (/ NaChar (+ 1.0 (exp (/ Vef KbT))))
          (/ NdChar (- -1.0 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT)))))))
   (if (<= Vef -4.5e+123)
     t_0
     (if (<= Vef 1.86e+71)
       (-
        (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept Ev) mu) KbT))))
        (/ NdChar (- -1.0 (exp (/ (- (+ mu EDonor) Ec) KbT)))))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar / (1.0 + exp((Vef / KbT)))) - (NdChar / (-1.0 - exp(((mu - ((Ec - Vef) - EDonor)) / KbT))));
	double tmp;
	if (Vef <= -4.5e+123) {
		tmp = t_0;
	} else if (Vef <= 1.86e+71) {
		tmp = (NaChar / (1.0 + exp((((EAccept + Ev) - mu) / KbT)))) - (NdChar / (-1.0 - exp((((mu + EDonor) - Ec) / KbT))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (nachar / (1.0d0 + exp((vef / kbt)))) - (ndchar / ((-1.0d0) - exp(((mu - ((ec - vef) - edonor)) / kbt))))
    if (vef <= (-4.5d+123)) then
        tmp = t_0
    else if (vef <= 1.86d+71) then
        tmp = (nachar / (1.0d0 + exp((((eaccept + ev) - mu) / kbt)))) - (ndchar / ((-1.0d0) - exp((((mu + edonor) - ec) / kbt))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar / (1.0 + Math.exp((Vef / KbT)))) - (NdChar / (-1.0 - Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT))));
	double tmp;
	if (Vef <= -4.5e+123) {
		tmp = t_0;
	} else if (Vef <= 1.86e+71) {
		tmp = (NaChar / (1.0 + Math.exp((((EAccept + Ev) - mu) / KbT)))) - (NdChar / (-1.0 - Math.exp((((mu + EDonor) - Ec) / KbT))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar / (1.0 + math.exp((Vef / KbT)))) - (NdChar / (-1.0 - math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT))))
	tmp = 0
	if Vef <= -4.5e+123:
		tmp = t_0
	elif Vef <= 1.86e+71:
		tmp = (NaChar / (1.0 + math.exp((((EAccept + Ev) - mu) / KbT)))) - (NdChar / (-1.0 - math.exp((((mu + EDonor) - Ec) / KbT))))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Vef / KbT)))) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)))))
	tmp = 0.0
	if (Vef <= -4.5e+123)
		tmp = t_0;
	elseif (Vef <= 1.86e+71)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Ev) - mu) / KbT)))) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(Float64(mu + EDonor) - Ec) / KbT)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NaChar / (1.0 + exp((Vef / KbT)))) - (NdChar / (-1.0 - exp(((mu - ((Ec - Vef) - EDonor)) / KbT))));
	tmp = 0.0;
	if (Vef <= -4.5e+123)
		tmp = t_0;
	elseif (Vef <= 1.86e+71)
		tmp = (NaChar / (1.0 + exp((((EAccept + Ev) - mu) / KbT)))) - (NdChar / (-1.0 - exp((((mu + EDonor) - Ec) / KbT))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -4.5e+123], t$95$0, If[LessEqual[Vef, 1.86e+71], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + Ev), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(N[(mu + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if Vef < -4.49999999999999983e123 or 1.86000000000000006e71 < Vef

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in Vef around inf

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 88.1

         \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   5. Applied rewrites 88.1%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   if -4.49999999999999983e123 < Vef < 1.86000000000000006e71

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in Vef around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}}} + \frac{NdChar}{1 + e^{\frac{\left(EDonor + mu\right) - Ec}{KbT}}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + mu\right) - Ec}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + mu\right) - Ec}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + mu\right) - Ec}{KbT}}}} + \frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}}} \]

      4. +-commutative N/A

         \[\leadsto \frac{NdChar}{\color{blue}{e^{\frac{\left(EDonor + mu\right) - Ec}{KbT}} + 1}} + \frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{NdChar}{\color{blue}{e^{\frac{\left(EDonor + mu\right) - Ec}{KbT}} + 1}} + \frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}}} \]

      6. lower-exp.f64 N/A

         \[\leadsto \frac{NdChar}{\color{blue}{e^{\frac{\left(EDonor + mu\right) - Ec}{KbT}}} + 1} + \frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{NdChar}{e^{\color{blue}{\frac{\left(EDonor + mu\right) - Ec}{KbT}}} + 1} + \frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}}} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{NdChar}{e^{\frac{\color{blue}{\left(EDonor + mu\right) - Ec}}{KbT}} + 1} + \frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{NdChar}{e^{\frac{\color{blue}{\left(mu + EDonor\right)} - Ec}{KbT}} + 1} + \frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}}} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\color{blue}{\left(mu + EDonor\right)} - Ec}{KbT}} + 1} + \frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}} + 1} + \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}}}} \]

      12. +-commutative N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}} + 1} + \frac{NaChar}{\color{blue}{e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}} + 1}} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}} + 1} + \frac{NaChar}{\color{blue}{e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}} + 1}} \]

   5. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\frac{NdChar}{e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}} + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -4.5 \cdot 10^{+123}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ \mathbf{elif}\;Vef \leq 1.86 \cdot 10^{+71}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + Ev\right) - mu}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{\left(mu + EDonor\right) - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{-1 - e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ \mathbf{if}\;Ev \leq -3.5 \cdot 10^{+115}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} - t\_0\\ \mathbf{elif}\;Ev \leq 8.3 \cdot 10^{-277}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (- -1.0 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT))))))
   (if (<= Ev -3.5e+115)
     (- (/ NaChar (+ 1.0 (exp (/ Ev KbT)))) t_0)
     (if (<= Ev 8.3e-277)
       (- (/ NaChar (+ 1.0 (exp (/ Vef KbT)))) t_0)
       (- (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (-1.0 - exp(((mu - ((Ec - Vef) - EDonor)) / KbT)));
	double tmp;
	if (Ev <= -3.5e+115) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) - t_0;
	} else if (Ev <= 8.3e-277) {
		tmp = (NaChar / (1.0 + exp((Vef / KbT)))) - t_0;
	} else {
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ndchar / ((-1.0d0) - exp(((mu - ((ec - vef) - edonor)) / kbt)))
    if (ev <= (-3.5d+115)) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) - t_0
    else if (ev <= 8.3d-277) then
        tmp = (nachar / (1.0d0 + exp((vef / kbt)))) - t_0
    else
        tmp = (nachar / (1.0d0 + exp((eaccept / kbt)))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (-1.0 - Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)));
	double tmp;
	if (Ev <= -3.5e+115) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) - t_0;
	} else if (Ev <= 8.3e-277) {
		tmp = (NaChar / (1.0 + Math.exp((Vef / KbT)))) - t_0;
	} else {
		tmp = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) - t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (-1.0 - math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))
	tmp = 0
	if Ev <= -3.5e+115:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) - t_0
	elif Ev <= 8.3e-277:
		tmp = (NaChar / (1.0 + math.exp((Vef / KbT)))) - t_0
	else:
		tmp = (NaChar / (1.0 + math.exp((EAccept / KbT)))) - t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT))))
	tmp = 0.0
	if (Ev <= -3.5e+115)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) - t_0);
	elseif (Ev <= 8.3e-277)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Vef / KbT)))) - t_0);
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (-1.0 - exp(((mu - ((Ec - Vef) - EDonor)) / KbT)));
	tmp = 0.0;
	if (Ev <= -3.5e+115)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) - t_0;
	elseif (Ev <= 8.3e-277)
		tmp = (NaChar / (1.0 + exp((Vef / KbT)))) - t_0;
	else
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(-1.0 - N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Ev, -3.5e+115], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[Ev, 8.3e-277], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if Ev < -3.50000000000000005e115

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in Ev around inf

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{Ev}{KbT}}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 94.6

         \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{Ev}{KbT}}}} \]

   5. Applied rewrites 94.6%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{Ev}{KbT}}}} \]

   if -3.50000000000000005e115 < Ev < 8.29999999999999978e-277

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in Vef around inf

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 76.2

         \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   5. Applied rewrites 76.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   if 8.29999999999999978e-277 < Ev

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in EAccept around inf

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 68.7

         \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]

   5. Applied rewrites 68.7%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Ev \leq -3.5 \cdot 10^{+115}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ \mathbf{elif}\;Ev \leq 8.3 \cdot 10^{-277}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (-
  (/ NdChar (+ 1.0 (exp (/ (- mu (- (- Ec Vef) EDonor)) KbT))))
  (/ NaChar (- -1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    code = (ndchar / (1.0d0 + exp(((mu - ((ec - vef) - edonor)) / kbt)))) - (nachar / ((-1.0d0) - exp((((eaccept + (ev + vef)) - mu) / kbt))))
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NdChar / (1.0 + Math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return (NdChar / (1.0 + math.exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - math.exp((((EAccept + (Ev + Vef)) - mu) / KbT))))

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu - Float64(Float64(Ec - Vef) - EDonor)) / KbT)))) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)))))
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = (NdChar / (1.0 + exp(((mu - ((Ec - Vef) - EDonor)) / KbT)))) - (NaChar / (-1.0 - exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu - N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(\left(Ec - Vef\right) - EDonor\right)}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} \]
4. Add Preprocessing

Alternative 17: 68.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(mu + Vef\right) + EDonor\\ t_1 := \frac{NdChar}{1 + e^{\frac{t\_0 - Ec}{KbT}}}\\ \mathbf{if}\;NdChar \leq -8 \cdot 10^{-23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NdChar \leq 9.2 \cdot 10^{-154}:\\ \;\;\;\;\frac{NdChar}{\mathsf{fma}\left(\frac{Ec - t\_0}{KbT}, -1, 2\right)} + \frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (+ mu Vef) EDonor))
        (t_1 (/ NdChar (+ 1.0 (exp (/ (- t_0 Ec) KbT))))))
   (if (<= NdChar -8e-23)
     t_1
     (if (<= NdChar 9.2e-154)
       (+
        (/ NdChar (fma (/ (- Ec t_0) KbT) -1.0 2.0))
        (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT)))))
       t_1))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (mu + Vef) + EDonor;
	double t_1 = NdChar / (1.0 + exp(((t_0 - Ec) / KbT)));
	double tmp;
	if (NdChar <= -8e-23) {
		tmp = t_1;
	} else if (NdChar <= 9.2e-154) {
		tmp = (NdChar / fma(((Ec - t_0) / KbT), -1.0, 2.0)) + (NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(mu + Vef) + EDonor)
	t_1 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(t_0 - Ec) / KbT))))
	tmp = 0.0
	if (NdChar <= -8e-23)
		tmp = t_1;
	elseif (NdChar <= 9.2e-154)
		tmp = Float64(Float64(NdChar / fma(Float64(Float64(Ec - t_0) / KbT), -1.0, 2.0)) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(1.0 + N[Exp[N[(N[(t$95$0 - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -8e-23], t$95$1, If[LessEqual[NdChar, 9.2e-154], N[(N[(NdChar / N[(N[(N[(Ec - t$95$0), $MachinePrecision] / KbT), $MachinePrecision] * -1.0 + 2.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -7.99999999999999968e-23 or 9.1999999999999999e-154 < NdChar

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 24.5

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 24.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

   6. Taylor expanded in NdChar around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NaChar} \]
   7. Step-by-step derivation

   8. Applied rewrites 12.6%

      \[\leadsto 0.5 \cdot \color{blue}{NaChar} \]

   9. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

       2. +-commutative N/A

          \[\leadsto \frac{NdChar}{\color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}} + 1}} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{NdChar}{\color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}} + 1}} \]

       4. lower-exp.f64 N/A

          \[\leadsto \frac{NdChar}{\color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}} + 1} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{NdChar}{e^{\color{blue}{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}} + 1} \]

       6. lower--.f64 N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\color{blue}{\left(EDonor + \left(Vef + mu\right)\right) - Ec}}{KbT}} + 1} \]

       7. +-commutative N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\color{blue}{\left(\left(Vef + mu\right) + EDonor\right)} - Ec}{KbT}} + 1} \]

       8. lower-+.f64 N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\color{blue}{\left(\left(Vef + mu\right) + EDonor\right)} - Ec}{KbT}} + 1} \]

       9. +-commutative N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\left(\color{blue}{\left(mu + Vef\right)} + EDonor\right) - Ec}{KbT}} + 1} \]

       10. lower-+.f64 71.1

           \[\leadsto \frac{NdChar}{e^{\frac{\left(\color{blue}{\left(mu + Vef\right)} + EDonor\right) - Ec}{KbT}} + 1} \]

   11. Applied rewrites 71.1%

       \[\leadsto \color{blue}{\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + 1}} \]

   if -7.99999999999999968e-23 < NdChar < 9.1999999999999999e-154

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \frac{NdChar}{1 + \color{blue}{e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. distribute-frac-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\mathsf{neg}\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. neg-mul-1 N/A

         \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{-1 \cdot \frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. exp-prod N/A

         \[\leadsto \frac{NdChar}{1 + \color{blue}{{\left(e^{-1}\right)}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{NdChar}{1 + \color{blue}{{\left(e^{-1}\right)}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{NdChar}{1 + {\color{blue}{\left(e^{-1}\right)}}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      9. lower-/.f64 100.0

         \[\leadsto \frac{NdChar}{1 + {\left(e^{-1}\right)}^{\color{blue}{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{{\left(e^{-1}\right)}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   5. Taylor expanded in KbT around inf

      \[\leadsto \frac{NdChar}{\color{blue}{2 + -1 \cdot \frac{Ec - \left(EDonor + \left(Vef + mu\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{NdChar}{\color{blue}{-1 \cdot \frac{Ec - \left(EDonor + \left(Vef + mu\right)\right)}{KbT} + 2}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{NdChar}{\color{blue}{\frac{Ec - \left(EDonor + \left(Vef + mu\right)\right)}{KbT} \cdot -1} + 2} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{NdChar}{\color{blue}{\mathsf{fma}\left(\frac{Ec - \left(EDonor + \left(Vef + mu\right)\right)}{KbT}, -1, 2\right)}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{NdChar}{\mathsf{fma}\left(\color{blue}{\frac{Ec - \left(EDonor + \left(Vef + mu\right)\right)}{KbT}}, -1, 2\right)} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{NdChar}{\mathsf{fma}\left(\frac{\color{blue}{Ec - \left(EDonor + \left(Vef + mu\right)\right)}}{KbT}, -1, 2\right)} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. +-commutative N/A

         \[\leadsto \frac{NdChar}{\mathsf{fma}\left(\frac{Ec - \color{blue}{\left(\left(Vef + mu\right) + EDonor\right)}}{KbT}, -1, 2\right)} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{NdChar}{\mathsf{fma}\left(\frac{Ec - \color{blue}{\left(\left(Vef + mu\right) + EDonor\right)}}{KbT}, -1, 2\right)} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      8. +-commutative N/A

         \[\leadsto \frac{NdChar}{\mathsf{fma}\left(\frac{Ec - \left(\color{blue}{\left(mu + Vef\right)} + EDonor\right)}{KbT}, -1, 2\right)} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      9. lower-+.f64 82.3

         \[\leadsto \frac{NdChar}{\mathsf{fma}\left(\frac{Ec - \left(\color{blue}{\left(mu + Vef\right)} + EDonor\right)}{KbT}, -1, 2\right)} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   7. Applied rewrites 82.3%

      \[\leadsto \frac{NdChar}{\color{blue}{\mathsf{fma}\left(\frac{Ec - \left(\left(mu + Vef\right) + EDonor\right)}{KbT}, -1, 2\right)}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -8 \cdot 10^{-23}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 9.2 \cdot 10^{-154}:\\ \;\;\;\;\frac{NdChar}{\mathsf{fma}\left(\frac{Ec - \left(\left(mu + Vef\right) + EDonor\right)}{KbT}, -1, 2\right)} + \frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 69.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{if}\;NaChar \leq -1.7 \cdot 10^{-97}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NaChar \leq 5.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept (+ Ev Vef)) mu) KbT))))))
   (if (<= NaChar -1.7e-97)
     t_0
     (if (<= NaChar 5.8e-65)
       (/ NdChar (+ 1.0 (exp (/ (- (+ (+ mu Vef) EDonor) Ec) KbT))))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	double tmp;
	if (NaChar <= -1.7e-97) {
		tmp = t_0;
	} else if (NaChar <= 5.8e-65) {
		tmp = NdChar / (1.0 + exp(((((mu + Vef) + EDonor) - Ec) / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp((((eaccept + (ev + vef)) - mu) / kbt)))
    if (nachar <= (-1.7d-97)) then
        tmp = t_0
    else if (nachar <= 5.8d-65) then
        tmp = ndchar / (1.0d0 + exp(((((mu + vef) + edonor) - ec) / kbt)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	double tmp;
	if (NaChar <= -1.7e-97) {
		tmp = t_0;
	} else if (NaChar <= 5.8e-65) {
		tmp = NdChar / (1.0 + Math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp((((EAccept + (Ev + Vef)) - mu) / KbT)))
	tmp = 0
	if NaChar <= -1.7e-97:
		tmp = t_0
	elif NaChar <= 5.8e-65:
		tmp = NdChar / (1.0 + math.exp(((((mu + Vef) + EDonor) - Ec) / KbT)))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT))))
	tmp = 0.0
	if (NaChar <= -1.7e-97)
		tmp = t_0;
	elseif (NaChar <= 5.8e-65)
		tmp = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp((((EAccept + (Ev + Vef)) - mu) / KbT)));
	tmp = 0.0;
	if (NaChar <= -1.7e-97)
		tmp = t_0;
	elseif (NaChar <= 5.8e-65)
		tmp = NdChar / (1.0 + exp(((((mu + Vef) + EDonor) - Ec) / KbT)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -1.7e-97], t$95$0, If[LessEqual[NaChar, 5.8e-65], N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -1.6999999999999999e-97 or 5.7999999999999996e-65 < NaChar

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{NaChar}{\color{blue}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{NaChar}{\color{blue}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1}} \]

      4. lower-exp.f64 N/A

         \[\leadsto \frac{NaChar}{\color{blue}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} + 1} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{NaChar}{e^{\color{blue}{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} + 1} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{NaChar}{e^{\frac{\color{blue}{\left(EAccept + \left(Ev + Vef\right)\right) - mu}}{KbT}} + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{NaChar}{e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right)} - mu}{KbT}} + 1} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{NaChar}{e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right)} - mu}{KbT}} + 1} \]

      9. lower-+.f64 70.0

         \[\leadsto \frac{NaChar}{e^{\frac{\left(\color{blue}{\left(Ev + Vef\right)} + EAccept\right) - mu}{KbT}} + 1} \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}} \]

   if -1.6999999999999999e-97 < NaChar < 5.7999999999999996e-65

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 31.5

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 31.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

   6. Taylor expanded in NdChar around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NaChar} \]
   7. Step-by-step derivation

   8. Applied rewrites 12.1%

      \[\leadsto 0.5 \cdot \color{blue}{NaChar} \]

   9. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]

       2. +-commutative N/A

          \[\leadsto \frac{NdChar}{\color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}} + 1}} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{NdChar}{\color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}} + 1}} \]

       4. lower-exp.f64 N/A

          \[\leadsto \frac{NdChar}{\color{blue}{e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}} + 1} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{NdChar}{e^{\color{blue}{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}} + 1} \]

       6. lower--.f64 N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\color{blue}{\left(EDonor + \left(Vef + mu\right)\right) - Ec}}{KbT}} + 1} \]

       7. +-commutative N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\color{blue}{\left(\left(Vef + mu\right) + EDonor\right)} - Ec}{KbT}} + 1} \]

       8. lower-+.f64 N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\color{blue}{\left(\left(Vef + mu\right) + EDonor\right)} - Ec}{KbT}} + 1} \]

       9. +-commutative N/A

          \[\leadsto \frac{NdChar}{e^{\frac{\left(\color{blue}{\left(mu + Vef\right)} + EDonor\right) - Ec}{KbT}} + 1} \]

       10. lower-+.f64 77.0

           \[\leadsto \frac{NdChar}{e^{\frac{\left(\color{blue}{\left(mu + Vef\right)} + EDonor\right) - Ec}{KbT}} + 1} \]

   11. Applied rewrites 77.0%

       \[\leadsto \color{blue}{\frac{NdChar}{e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}} + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -1.7 \cdot 10^{-97}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 5.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 62.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\\ t_1 := \left(NaChar + NdChar\right) \cdot 0.5\\ \mathbf{if}\;KbT \leq -2.8 \cdot 10^{+204}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \left(\frac{NdChar}{KbT} - \frac{NaChar}{KbT}\right) \cdot mu, t\_1\right)\\ \mathbf{elif}\;KbT \leq 2.6 \cdot 10^{+225}:\\ \;\;\;\;\frac{NaChar}{1 + e^{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \mathsf{fma}\left(NaChar, t\_0, \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT} \cdot NdChar\right), t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ (- (+ EAccept (+ Ev Vef)) mu) KbT))
        (t_1 (* (+ NaChar NdChar) 0.5)))
   (if (<= KbT -2.8e+204)
     (fma -0.25 (* (- (/ NdChar KbT) (/ NaChar KbT)) mu) t_1)
     (if (<= KbT 2.6e+225)
       (/ NaChar (+ 1.0 (exp t_0)))
       (fma
        -0.25
        (fma NaChar t_0 (* (/ (- (+ (+ mu Vef) EDonor) Ec) KbT) NdChar))
        t_1)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = ((EAccept + (Ev + Vef)) - mu) / KbT;
	double t_1 = (NaChar + NdChar) * 0.5;
	double tmp;
	if (KbT <= -2.8e+204) {
		tmp = fma(-0.25, (((NdChar / KbT) - (NaChar / KbT)) * mu), t_1);
	} else if (KbT <= 2.6e+225) {
		tmp = NaChar / (1.0 + exp(t_0));
	} else {
		tmp = fma(-0.25, fma(NaChar, t_0, (((((mu + Vef) + EDonor) - Ec) / KbT) * NdChar)), t_1);
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(Float64(EAccept + Float64(Ev + Vef)) - mu) / KbT)
	t_1 = Float64(Float64(NaChar + NdChar) * 0.5)
	tmp = 0.0
	if (KbT <= -2.8e+204)
		tmp = fma(-0.25, Float64(Float64(Float64(NdChar / KbT) - Float64(NaChar / KbT)) * mu), t_1);
	elseif (KbT <= 2.6e+225)
		tmp = Float64(NaChar / Float64(1.0 + exp(t_0)));
	else
		tmp = fma(-0.25, fma(NaChar, t_0, Float64(Float64(Float64(Float64(Float64(mu + Vef) + EDonor) - Ec) / KbT) * NdChar)), t_1);
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(N[(EAccept + N[(Ev + Vef), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar + NdChar), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[KbT, -2.8e+204], N[(-0.25 * N[(N[(N[(NdChar / KbT), $MachinePrecision] - N[(NaChar / KbT), $MachinePrecision]), $MachinePrecision] * mu), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[KbT, 2.6e+225], N[(NaChar / N[(1.0 + N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(NaChar * t$95$0 + N[(N[(N[(N[(N[(mu + Vef), $MachinePrecision] + EDonor), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision] * NdChar), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -2.80000000000000025e204

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right) + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(\frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right)} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}, \frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)} \]

   5. Applied rewrites 76.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, \mathsf{fma}\left(NaChar, \frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}, NdChar \cdot \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}\right), 0.5 \cdot \left(NdChar + NaChar\right)\right)} \]

   6. Taylor expanded in EAccept around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, \frac{EAccept \cdot NaChar}{\color{blue}{KbT}}, \frac{1}{2} \cdot \left(NdChar + NaChar\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 76.2%

      \[\leadsto \mathsf{fma}\left(-0.25, EAccept \cdot \color{blue}{\frac{NaChar}{KbT}}, 0.5 \cdot \left(NdChar + NaChar\right)\right) \]

   9. Taylor expanded in mu around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, mu \cdot \color{blue}{\left(-1 \cdot \frac{NaChar}{KbT} + \frac{NdChar}{KbT}\right)}, \frac{1}{2} \cdot \left(NdChar + NaChar\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 80.2%

       \[\leadsto \mathsf{fma}\left(-0.25, \left(\frac{NdChar}{KbT} - \frac{NaChar}{KbT}\right) \cdot \color{blue}{mu}, 0.5 \cdot \left(NdChar + NaChar\right)\right) \]

   if -2.80000000000000025e204 < KbT < 2.60000000000000004e225

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in NdChar around 0

      \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{NaChar}{\color{blue}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{NaChar}{\color{blue}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}} + 1}} \]

      4. lower-exp.f64 N/A

         \[\leadsto \frac{NaChar}{\color{blue}{e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} + 1} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{NaChar}{e^{\color{blue}{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}} + 1} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{NaChar}{e^{\frac{\color{blue}{\left(EAccept + \left(Ev + Vef\right)\right) - mu}}{KbT}} + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{NaChar}{e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right)} - mu}{KbT}} + 1} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{NaChar}{e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right)} - mu}{KbT}} + 1} \]

      9. lower-+.f64 63.0

         \[\leadsto \frac{NaChar}{e^{\frac{\left(\color{blue}{\left(Ev + Vef\right)} + EAccept\right) - mu}{KbT}} + 1} \]

   5. Applied rewrites 63.0%

      \[\leadsto \color{blue}{\frac{NaChar}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}} + 1}} \]

   if 2.60000000000000004e225 < KbT

   1. Initial program 99.8%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right) + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(\frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right)} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}, \frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)} \]

   5. Applied rewrites 83.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, \mathsf{fma}\left(NaChar, \frac{\left(\left(Ev + Vef\right) + EAccept\right) - mu}{KbT}, NdChar \cdot \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT}\right), 0.5 \cdot \left(NdChar + NaChar\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -2.8 \cdot 10^{+204}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \left(\frac{NdChar}{KbT} - \frac{NaChar}{KbT}\right) \cdot mu, \left(NaChar + NdChar\right) \cdot 0.5\right)\\ \mathbf{elif}\;KbT \leq 2.6 \cdot 10^{+225}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \mathsf{fma}\left(NaChar, \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}, \frac{\left(\left(mu + Vef\right) + EDonor\right) - Ec}{KbT} \cdot NdChar\right), \left(NaChar + NdChar\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 22.7% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -1.65 \cdot 10^{-97}:\\ \;\;\;\;0.5 \cdot NaChar\\ \mathbf{elif}\;NaChar \leq 2.7 \cdot 10^{+91}:\\ \;\;\;\;0.5 \cdot NdChar\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot NaChar\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NaChar -1.65e-97)
   (* 0.5 NaChar)
   (if (<= NaChar 2.7e+91) (* 0.5 NdChar) (* 0.5 NaChar))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -1.65e-97) {
		tmp = 0.5 * NaChar;
	} else if (NaChar <= 2.7e+91) {
		tmp = 0.5 * NdChar;
	} else {
		tmp = 0.5 * NaChar;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (nachar <= (-1.65d-97)) then
        tmp = 0.5d0 * nachar
    else if (nachar <= 2.7d+91) then
        tmp = 0.5d0 * ndchar
    else
        tmp = 0.5d0 * nachar
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -1.65e-97) {
		tmp = 0.5 * NaChar;
	} else if (NaChar <= 2.7e+91) {
		tmp = 0.5 * NdChar;
	} else {
		tmp = 0.5 * NaChar;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NaChar <= -1.65e-97:
		tmp = 0.5 * NaChar
	elif NaChar <= 2.7e+91:
		tmp = 0.5 * NdChar
	else:
		tmp = 0.5 * NaChar
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NaChar <= -1.65e-97)
		tmp = Float64(0.5 * NaChar);
	elseif (NaChar <= 2.7e+91)
		tmp = Float64(0.5 * NdChar);
	else
		tmp = Float64(0.5 * NaChar);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NaChar <= -1.65e-97)
		tmp = 0.5 * NaChar;
	elseif (NaChar <= 2.7e+91)
		tmp = 0.5 * NdChar;
	else
		tmp = 0.5 * NaChar;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NaChar, -1.65e-97], N[(0.5 * NaChar), $MachinePrecision], If[LessEqual[NaChar, 2.7e+91], N[(0.5 * NdChar), $MachinePrecision], N[(0.5 * NaChar), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -1.6500000000000001e-97 or 2.7e91 < NaChar

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 25.9

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 25.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

   6. Taylor expanded in NdChar around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NaChar} \]
   7. Step-by-step derivation

   8. Applied rewrites 22.3%

      \[\leadsto 0.5 \cdot \color{blue}{NaChar} \]

   if -1.6500000000000001e-97 < NaChar < 2.7e91

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing

   3. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

      4. lower-+.f64 28.1

         \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

   5. Applied rewrites 28.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

   6. Taylor expanded in NdChar around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NaChar} \]
   7. Step-by-step derivation

   8. Applied rewrites 12.8%

      \[\leadsto 0.5 \cdot \color{blue}{NaChar} \]

   9. Taylor expanded in NdChar around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{NdChar} \]
   10. Step-by-step derivation

   11. Applied rewrites 26.0%

       \[\leadsto 0.5 \cdot \color{blue}{NdChar} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 27.7% accurate, 30.7× speedup

Math

\[\begin{array}{l} \\ \left(NaChar + NdChar\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (* (+ NaChar NdChar) 0.5))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NaChar + NdChar) * 0.5;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    code = (nachar + ndchar) * 0.5d0
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NaChar + NdChar) * 0.5;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return (NaChar + NdChar) * 0.5

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(Float64(NaChar + NdChar) * 0.5)
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = (NaChar + NdChar) * 0.5;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(N[(NaChar + NdChar), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
2. Add Preprocessing

3. Taylor expanded in KbT around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

   2. distribute-lft-out N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

   4. lower-+.f64 27.1

      \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

5. Applied rewrites 27.1%

   \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

6. Final simplification 27.1%

   \[\leadsto \left(NaChar + NdChar\right) \cdot 0.5 \]
7. Add Preprocessing

Alternative 22: 17.8% accurate, 46.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot NaChar \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (* 0.5 NaChar))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return 0.5 * NaChar;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    code = 0.5d0 * nachar
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return 0.5 * NaChar;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return 0.5 * NaChar

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(0.5 * NaChar)
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.5 * NaChar;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(0.5 * NaChar), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
2. Add Preprocessing

3. Taylor expanded in KbT around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar + \frac{1}{2} \cdot NaChar} \]

   2. distribute-lft-out N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(NdChar + NaChar\right)} \]

   4. lower-+.f64 27.1

      \[\leadsto 0.5 \cdot \color{blue}{\left(NdChar + NaChar\right)} \]

5. Applied rewrites 27.1%

   \[\leadsto \color{blue}{0.5 \cdot \left(NdChar + NaChar\right)} \]

6. Taylor expanded in NdChar around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{NaChar} \]
7. Step-by-step derivation

8. Applied rewrites 17.1%

   \[\leadsto 0.5 \cdot \color{blue}{NaChar} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024331 
(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
  :name "Bulmash initializePoisson"
  :precision binary64
  (+ (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT)))) (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT))))))